Geometry of non-commutative orbits related to Hecke symmetries

To some braiding R of Hecke type (a Hecke symmetry) we put into correspondence an associative algebra called the modified Reflection Equation Algebra (mREA). We construct a series of matrices L(m), m = 1, 2, ... with entries belonging to such an algebra so that each of them satisfies a version of the Cayley-Hamilton identity with central coefficients. We also consider some quotients of the mREA which are called the non-commutative orbits. For each of these orbits we construct a large family of projective modules. In such a family we introduce an algebraic structure which is close to that of K(Fl(C)). This algebraic structure respects an equivalence relation motivated by a ”quantum” trace compatible with the initial Hecke symmetry R. For a subclass of non-commutative orbits we compute the spectrum of central elements TrR L k (m), k ∈ N of the mREA AMS Mathematics Subject Classification, 1991: 17B37, 81R50